Locally modelled regression and functional data

Abstract

The general framework of this paper deals with the nonparametric regression of a scalar response on a functional variable (i.e. one observation can be a curve, surface, or any other object lying into an infinite-dimensional space). This paper proposes to model local behaviour of the regression operator (i.e. the link between a scalar response and an explanatory functional variable). To this end, one introduces a functional approach in the same spirit as local linear ideas in nonparametric regression. The main advantage of this functional local method is to propose an explicit expression of a kernel-type estimator which makes its computation easy and fast while keeping good predictive performance. Asymptotic properties are stated, and a functional data set illustrates the good behaviour of this functional locally modelled regression method.